Sharp constants for weighted Moser-Trudinger inequalities on groups of Heisenberg type

Let G be a group of Heisenberg type, Q=m+2q be its homogeneous dimension, Qa=Q−a, Qa′=Q−aQ−a−1. For u∈G, we write u=(z(u),t(u))∈G, where t(u) is the coordinate of u corresponding to the center T of the Lie algebra G of G, z(u) is corresponding to the orthogonal complement of T. Let N(u)=(|z(u)|4+t(u)2)14 be the homogeneous norm of u∈G, W(u)=|z(u)|−a be a weight. The main purpose of this paper is to establish sharp constants for weighted Moser-Trudinger inequalities on domains of finite measure in G (Theorem 2.1) and on unbounded domains (Theorem 2.2). We also establish the weighted inequalities of Adachi-Tanaka type on the entire G (Theorem 2.3). Our results extend the sharp Moser-Trudinger inequalities on domains of finite measure in Cohn and Lu (2001, 2002) [13], [14] and on unbounded domains in Lam et al. (2012) [19] to the weighted case and improve the sharp weighted Moser-Trudinger inequality proved in Tyson (2006) [16] on domains of finite measure on G. The usual symmetrization method (i.e., rearrangement argument) is not available on such groups and therefore our argument is a rearrangement-free argument recently developed in Lam and Lu (2012)  [17], [18]. Our weighted Adachi-Tanaka type inequalities extend the nonweighted results in Lam et al. (2012)  [20].